Analysis of RT distributions with R

1. Analysis of RT distributionswith R Emil Ratko-Dehnert WS 2010/ 2011 Session 10 – 25.01.2011

2. Last time... • RT distributions in the field • Convolution • Ex-Gaussian, Ex-Wald, Gamma, Weibull • Comparing functional fits • Bootstrapping • Creating functions in R

3. Ex-Gauss Ex-Gauss distribution • Convolution of an exponential and a gaussian distribution • Correspondance to mental processes (exp -> decision, gauss -> residual perceptual/ response processes) • Very good fits to RT data

4. Ex-Wald Ex-Wald distribution • Is the convolution of a Wald and an exponential distribution • Represents decision and response components as a diffusion process (Schwarz, 2001) • Neurally plausible (single cell recordings) + parameters can be interpreted psychologically • Very good fits to RT data and highly successful in modelling RTs in various cognitive fields

5. Ex-Wald Diffusion Process Information space Respond „A“ A Mean drift ν z time Boundary separation drift rate ~N(ν,η) Evidence B Respond „B“

6. Gamma Gamma distribution • Series of exponential distributions • α = average scale of processes, β = reflects approximate number of processes • Above average fits • Suitable for three-stage exponential models

7. Weibull Weibull Distribution • Like a series of races the weibull distribution renders an asymptotic description of their minima • γshould lie between 1 (exp.) and 3.6 (gauss.) • Decent functional fits, appropriate for processes which can be modelled as races

8. Palmer et al. (2009) • Compared functional fits for three different search tasks (feature, conjunction, spatial config.) • H0: fit to normal distribution • All proposed distributions could reject H0, but not equally well 1: Ex-Gauss, 2: Ex-Wald, 3: Gamma, 4:Weibull

9. II Functional Forms of Random variables

10. So far... • We looked at densities and (cumulative) distribution functions for analysis of RTs • As all densities for RTs are unimodal and rightskewed they can be inappropriate for analysis • Similarly all CDF are sigmoidal, so they might not be adequate to compare

11. RTs in ms

12. Survivor function • The survivor function F(t) is the probability that the lifetime of an object is at least t • In oder words: the probability that failure occurs after t

14. Hazard function • The hazard function h(t) gives the likelihood that an event will occur in the next small interval dt in time, given that it has not occured before that point in time • Thus, it is the conditional probability:

15. Connection to survivor function • When F(t) is differentiable, the hazard function can be expressed as a function of the survivor function F(t):

16. Ex: h(t) of Ex-Gauss and Wald

18. Cumulative Hazard Function • Accumulted hazard over time • Is an alternative (but equivalent) represen-tation of the hazard h(t) • cf. Density <-> Distribution

20. Literature • Ashby, Tein & Balakrishnan, 1993 • Bloxom, 1984 • Colonius, 1988 • Luce, 1986 • Maddox, Ashby & Gottlob, 1998 • Thomas, 1971 • Burbeck & Luce, 1982

21. III Estimation Theory

22. Next steps • Theoretical analysis of distributions and their discrimination is important but in research practice another aspect also paramount • „good“ estimation of densities, distribution and hazard functions are the first step to analyse RT data

23. And now to